BAR-CODES OF SIERPIŃSKI RELATIVES WITH TRIANGLE CONVEX HULLS

Abstract

This paper presents results about the triangle Sierpiński relatives. These are Sierpiński relatives that have the same convex hull (whose boundary is a right isosceles triangle) as the Sierpiński gasket. In general, the Sierpiński relatives all have the same fractal dimension but different topologies. The special subset of triangle relatives includes are all both path-connected and multiply-connected. We investigate the epsilon hulls of the relatives (sets of all points within a distance of [Formula: see text] to the relative) to characterize and compare. This analysis includes the topological bar-codes which convey information about the connectivity of the [Formula: see text]-hulls of the relatives as [Formula: see text] ranges over the non-negative reals. We also prove that the growth rate of holes in the [Formula: see text]-hulls is equal to the fractal dimension.